An effective p-adic analogue of a theorem of Thue II. The greatest prime factor of a binary form
An extension of Sophie Germain's method to a wide class of diophantine equations.
An identity Ramanujan probably missed
"Ramanujan's 6-10-8 identity" inspired Hirschhorn to formulate his "3-7-5 identity". Now, we give a new "6-14-10 identity" which we suppose Ramanujan would have discovered but missed to mention in his notebooks.
An note on the Tarry-Escott problem.
Applications of Jacobi's symbol to Lehmer's numbers
Arithmetic of elliptic curves and diophantine equations
We give a survey of methods used to connect the study of ternary diophantine equations to modern techniques coming from the theory of modular forms.
Arithmetic of Pell surfaces
Arithmetische Studien über den "letzten" Fermatschen Satz, welcher aussagt, dass die Gleichung an = bn + cn für n > 2 in ganzen Zahlen nicht auflösbar ist.
Arithmetische Studien üeber den "letzten" Fermatschen Satz, welcher aussagt, dass die Gleichung an=bn+cn für n>2 in ganzen Zahlen nicht auflösbar ist [Book]
Artin's conjecture for septic and unidecic forms
Asymptotic aspects of the Diophantine equation
Asymptotic growth of Markoff-Hurwitz numbers
Baker's explicit abc-conjecture and applications
Bernoulli numbers, Hurwitz numbers, p-adic L-functions and Kummer's criterion.
Let K = Q(ζp) and let hp be its class number. Kummer showed that p divides hp if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.
Binomial Thue's equation over function fields
Bounds for the minimal solution of genus zero diophantine equations
Bounds for the smallest integer point of a rational curve
Complete solutions of a Lebesgue-Ramanujan-Nagell type equation
We consider the Lebesgue-Ramanujan-Nagell type equation , where and are unknown integers with . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all -integral points on a class of elliptic curves.
Congruent numbers with higher exponents
This paper investigates the system of equations in positive integers , , , , where and are positive integers with . In case of we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient with the condition . Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero .
Constructing fifteen infinite classes of nonregular bipartite integral graphs.